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Manipal University Department of Chemical Engineering CE 1833- PROCESS DYNAMICS AND CONTROL LABORATORY IV YEAR, VIIITH SEMESTER, 2019 EXPERIMENT No. #4 Interacting Tanks INSTRUCTORS: Dr. Abhishek Sharma Dr. Anees Y. Khan GROUP #A Member name: Akriti Agarwal Reg. number: 159110003 Experiment carried on: February 5, 2019 Report submitted on: February 12, 2019
PRELAB (10) ______ EXECUTIVE SUMMARY (10) ______ INTRODUCTION/OBJECTIVES/SCOPE/PROCEDURE (30) ______ RESULTS & DISCUSSION (30) ______ a) Data analysis and interpretation of information b) Presentation of relevant information (including results on graphical, tabular or equation forms) CONCLUSIONS (5) ______ REFERENCES (5) ______ APPENDIX a) Original data, sample calculations, Excel sheets (5) ______ GENERAL COMPLETENESS a) Conciseness and neatness (5) ______ TOTAL (100) ______
1
2
Table of Contents 4.1 Executive Summary ................................................................................................................................ 4 4.2 Objective ................................................................................................................................................. 5 4.3 Introduction ............................................................................................................................................. 5 4.4 Experimental Set-up................................................................................................................................ 6 4.5
Procedure .......................................................................................................................................... 7
(a)
For a step change: ......................................................................................................................... 8
(b)
For an impulse change: ................................................................................................................. 8
4.6 Result and Discussion ............................................................................................................................. 8 a) For a step change .................................................................................................................................. 8 b) For an impulse change .......................................................................................................................... 9 4.7
Conclusion ...................................................................................................................................... 10
4.8 References ............................................................................................................................................. 10 4.9 Appendix A4 ......................................................................................................................................... 11
List of Figures Figure 1: Two tank interacting system ......................................................................................................... 5 Figure 2: Experimental Set-up ...................................................................................................................... 7 Figure 3: Response observed for a step change ............................................................................................ 9 Figure 4: Response to an impulse change ................................................................................................... 10
List of Tables Table 1: Observation Table for step change ............................................................................................... 11 Table 2: Observation for impulse change ................................................................................................... 14
3
4.1 Executive Summary The objective of the experiment is to study the dynamic response of two interacting tank system subjected to a step and impulse change. To achieve the objective two tanks each of area 66.4 cm2 was used. For a step change, the initial flowrate was set to 10LPH and the initial steady state height of tank 3 and 2 were obtained at 23cm and 14cm respectively and then a step change of 50LPH was introduced. The time constant of tank 3 and 2 were found to be 500s and 435s respectively. The observed and calculated height of tank 2 was found to be slightly different and the reason for the same may be parallax error or the non-linearity of resistance. For an impulse change the initial flowrate was set to 20LPH the initial steady state height of tank 3 and 2 were obtained at 24cm and 18cm respectively and then 500ml of water was added to 3rd tank and it was observed that the water level increases all of a sudden and then it starts to fall and reaches its initial steady state value after sometime.
4
4.2 Objective The objective of the experiment is to study the dynamic response of two interacting tank system subjected to step and impulse change
4.3 Introduction A physical system can be represented by several first-order processes connected in series [1]. One possible example for such an arrangement is shown in Fig.1.It consists of two tanks connected in an interacting manner in which two tanks are arranged such that the outlet flow from the first tank is the inlet flow to the second tank. In such a system the outlet flow from tank 1 discharges directly into tank 2, and the flow through R1 (flow resistance) depends on the difference between h1 and h2 (liquid level in tank 1 and 2 respectively).
Figure 1: Two tank interacting system [1]
Let us assume the liquid to be of constant density (Ο), the tanks to have uniform cross-sectional area (A1 and A2), and the flow resistances are R1 and R2. Using the notations mentioned on figure 1, we can write the following equations [1]: For Tank 1: πβ1 β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.1) ππ‘ β1 β β2 π 1 = β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.2) π1
π β π1 = π΄1
For Tank 2: π1 β π2 = π΄2
πβ2 β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.3) ππ‘
5
π 2 =
β2 β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.4) π2
At steady state the above equations can be written as: dh1s β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.5) dt h1s β h2s = β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.6) q1s
q s β q1s = A1 R1s
q1s β q 2s = A2 R 2s =
dh2s β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.7) dt
h2s β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.8) q 2s
On subtracting (4.1-4.5), (4.2-4.6), (4.3-4.7), (4.4-4.8), introducing deviation variables and taking Laplace Transforms we get: Q(s) β Q1 (s) = A1 (s)sH1 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ (4.9) Q1 (s) β Q2 (s) = A2 (s)sH2 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ (4.10) R1 (s)Q1 (s) = H1 (s) β H2 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.11) R 2 (s)Q2 (s) = H2 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.12) H2 (s) R2 = β¦ β¦ β¦ β¦ β¦ (4.13) Q(s) Ο2 Ο1 s 2 + (Ο1 + Ο2 + A1 R 2 )s + 1 The difference between the transfer function for the non-interacting system and that for the interacting system is the presence of the cross product term A1R2 in the coefficient of s.
4.4 Experimental Set-up The apparatus shown in Fig. 2 is used to study the response of a two tank non-interacting tanks in series. It consists of supply tank, pump for water circulation from the sump tank to either tank 1 or 3 depending upon the type of system to be studied, a rotameter for inlet flow measurement and 3 graduated tanks which can be connected in interacting and non-interacting mode. The
6
graduations in the tank help to measure the level of water in the tank at a particular time. For this experiment we use interacting mode i.e. tank 3 and tank 2.
Figure 2: Experimental Set-up
4.5 Procedure ο·
Ensure all drain valves are closed.
ο·
Fill water in the sump tank.
ο·
Switch on the pump and allow the water to flow to tank 3.
ο·
Set the desired flowrate of water to Tank 3 using the rotameter provided.
ο·
Adjust the valves so as to attain steady state in tank 3 and tank 2
ο·
Once steady state is achieved record the initial flowrate and the steady state level of both the tanks
7
(a) ο·
For a step change: Once the steady state is achieved, apply step up change by increasing the inlet flowrate to tank 3 using the rotameter.
ο·
Record the water level in both tanks after every fixed interval of time till steady state is reached.
ο·
Once the steady state is achieved, record the final level of both the tanks.
(b)
For an impulse change: ο·
Once the system reaches steady state apply impulse change by adding 500 ml of water directly to Tank 3 and record the first data immediately.
ο·
Record the water level in both tanks after every fixed interval of time till steady state is reached.
ο·
Once the steady state is achieved, record the final level of both the tanks.
4.6 Result and Discussion a) For a step change Two tanks each of diameter 9.2cm and area of 66.47cm2 was used in the experiment. The initial flowrate to tank 3 was maintained at 10LPH. Initial steady state level of water in tank 3 (h1s) was obtained at 22cm and initial steady state level of water in tank 2 (h2s) was found to be 14cm. A step up change of magnitude 50 was applied i.e. the flowrate was changed to 60LPH. The change in water level with respect to time, on applying step change, for both the tank is shown in Fig. 3. It also shows the calculated value of H2 and the calculation is shown in Appendix A4. The final water level obtained in tank 3 and tank 2 was 126 and 105 cm respectively. The values of Ο1 and Ο2 were calculated to be 500s and 435s respectively. The calculation is shown in Appendix A4. As it can be seen from Fig. 3 the observed and calculated value of H2(t) is not exactly same and the calculated response reaches steady state after a very long time as compared to what is observed. Deviations observed may be due to following factors: β’ Non-linearity of valve resistance. β’ Step change is not instantaneous. β’ Visual errors in recording observations. β’ Accuracy of rotameters. In terms of transient response the interacting system is more sluggish than the non-interacting system. 8
120 100
Height(cm)
80 H1
60
H2 observed
40
H2 calculated
20 0 0
100
200
300
400
500
600
700
time(s) Figure 3: Response observed for a step change
b) For an impulse change Two tanks each of diameter 9.2cm and area of 66.47cm2 was used in the experiment. The initial flowrate to tank 3 was maintained at 20LPH. Initial steady state level of water in tank 3 (h1s) was obtained at 24cm and initial steady state level of water in tank 2 (h2s) was found to be 18cm. For an impulse change 500cm3 of water was added to tank 3 quickly at the water level in tank 3 and 2 at t=0 was found to be 94cm and 42cm respectively. The change in water level with respect to time, on applying impulse change, for both the tank is shown in Fig. 4. The final water level obtained in tank 3 and tank 2 was 24cm and 18 cm respectively. It can be seen from Fig. 4 that the output for an impulse change first increases as soon as the impulse is applied and then decreases and reaches a value which is equal to the initial height. The curves obtained in Fig. 4 are not smooth. Deviations observed may be due to following factors: β’ Non-linearity of valve resistance. β’ Step change is not instantaneous. β’ Visual errors in recording observations. β’ Accuracy of rotameters.
9
80 70 60
height(cm)
50 40
H1
30
H2
20 10 0 0
50
100
150
200
250
300
350
400
450
-10 time(s) Figure 4: Response to an impulse change
4.7 Conclusion The objective of the experiment was achieved by using two cylindrical tanks of equal areas (66.4 cm2). For a step change of 50LPH in the flowrate the values of Ο1 and Ο2 obtained were 500s and 435s respectively. For an interacting tank system the flow through R1 depends on the difference in the water level in the two tanks. .For an impulse the water level first increases and then decreases and comes back to its original after sometime. In terms of transient response the interacting system is more sluggish than the non-interacting system. Any error that might have occurred may be due to parallax or non-linearity of valve resistance. Hence, for more accurate results, it is recommended that the experiment should be allowed to run for an hour before step or impulse change is introduced. This would ensure that steady state is achieved and hence would result in much more precise results.
4.8 References 1. D. Coughanowr and S. Le Blanc, Process System Analysis and Control, Mc-Graw Hill (1991)(71-92).
10
4.9 Appendix A4 Table 1: Observation Table for step change
time
h1
h2
H1
H2(observed) H2(calculated)
(s)
(cm)
(cm)
(cm)
(cm)
(cm)
0
22
14
0
0
0
10
32
15
10
1
0.07377779
20
37
15
15
1
0.289038299
30
41
15
19
1
0.637043998
40
46
15
24
1
1.109531867
50
49
16
27
2
1.698688095
60
53
16
31
2
2.397124126
70
56
17
34
3
3.197853975
80
60
18
38
4
4.094272759
90
64
19
42
5
5.080136361
100
66
20
44
6
6.149542177
110
70
21
48
7
7.296910889
120
72
22
50
8
8.516969206
130
75
23
53
9
9.804733522
140
78
24
56
10
11.15549445
150
82
26
60
12
12.56480215
160
84
27
62
13
14.0284525
170
86
28
64
14
15.54247392
180
88
30
66
16
17.10311496
190
90
32
68
18
18.70683254
200
92
34
70
20
20.35028076
210
94
36
72
22
22.03030039
220
96
37
74
23
23.74390885
230
98
39
76
25
25.48829078
240
100
41
78
27
27.26078905
11
250
101
43
79
29
29.05889628
260
102
45
80
31
30.88024684
270
103
48
81
34
32.72260923
280
104
50
82
36
34.58387886
290
105
53
83
39
36.46207128
300
106
55
84
41
38.3553157
310
107
57
85
43
40.26184889
320
108
60
86
46
42.18000937
330
109
62
87
48
44.10823198
340
110
64
88
50
46.04504267
350
111
66
89
52
47.9890536
360
112
67
90
53
49.93895846
370
113
68
91
54
51.89352815
380
114
69
92
55
53.8516065
390
115
70
93
56
55.81210646
400
116
71
94
57
57.77400624
410
117
73
95
59
59.73634587
420
118
75
96
61
61.6982238
430
119
77
97
63
63.65879376
440
120
79
98
65
65.61726176
450
120
81
98
67
67.57288322
460
120
83
98
69
69.52496032
470
120
86
98
72
71.47283945
480
120
89
98
75
73.41590878
490
121
92
99
78
75.353596
500
121
94
99
80
77.28536614
510
121
95
99
81
79.21071958
520
121
96
99
82
81.12919003
530
122
97
100
83
83.04034277
540
122
98
100
84
84.94377291
12
550
123
99
101
85
86.83910371
560
123
100
101
86
88.72598505
570
124
101
102
87
90.60409197
580
124
103
102
89
92.47312328
590
125
103
103
89
94.3328002
600
125
104
103
90
96.18286518
610
126
104
104
90
98.02308067
620
126
104
104
90
99.85322801
630
126
105
104
91
101.6731064
640
126
105
104
91
103.4825319
650
126
105
104
91
105.2813364
660
126
105
104
91
107.0693668
Calculation: For time constant of Tank 1 dQ1 = 60 - 10 = 50 LPH dH1 = 126 - 22 = 104cm R1 = dH1/ dQ1 = 7.488 s/cm2 Ο1 = A1*R1 = 500s For calculated height of tank 2: π=
1 1 π΄1 π 2 + + = 0.006302 π1 π2 π1 π2
π π 1 πΌ = (β ) + β( )2 β = β0.00084 2 2 π1 π2 π π 1 π½ = (β ) β β( )2 β = β0.00546 2 2 π1 π2 1 1 [πΌ π πΌπ‘ ] β π π½π‘ π½ π»2 (π‘)πππππ’πππ‘ππ = π΄π 2 {1 β } 1 1 πΌβπ½ At t=10 secs H2=0.074 cm. 13
Table 2: Observation for impulse change
time
h1
h2
H1
H2
Steady
24
18
0
0
0
94
42
70
24
10
74
44
50
26
15
70
46
46
28
20
66
48
42
30
30
60
50
36
32
40
58
48
34
30
50
57
46
33
28
60
56
44
32
26
70
54
42
30
24
80
52
40
28
22
90
50
38
26
20
100
49
36
25
18
110
47
34
23
16
120
46
33
22
15
130
45
32
21
14
140
44
31
20
13
150
43
30
19
12
160
42
30
18
12
170
41
29
17
11
180
40
28
16
10
190
39
27
15
9
200
38
26
14
8
210
37
25
13
7
220
36
25
12
7
230
35
24
11
6
240
34
23
10
5
state
14
250
33
22
9
4
260
32
21
8
3
270
31
20
7
2
280
30
20
6
2
290
30
19
6
1
300
29
19
5
1
310
28
18
4
0
320
28
18
4
0
330
27
18
3
0
340
27
18
3
0
350
26
18
2
0
360
26
18
2
0
370
25
18
1
0
380
25
18
1
0
390
24
18
0
0
400
24
18
0
0
410
24
18
0
0
420
24
18
0
0
15
PRELAB (10) ______ EXECUTIVE SUMMARY (10) ______ INTRODUCTION/OBJECTIVES/SCOPE/PROCEDURE (30) ______ RESULTS & DISCUSSION (30) ______ a) Data analysis and interpretation of information b) Presentation of relevant information (including results on graphical, tabular or equation forms) CONCLUSIONS (5) ______ REFERENCES (5) ______ APPENDIX a) Original data, sample calculations, Excel sheets (5) ______ GENERAL COMPLETENESS a) Conciseness and neatness (5) ______ TOTAL (100) ______
1
2
Table of Contents 4.1 Executive Summary ................................................................................................................................ 4 4.2 Objective ................................................................................................................................................. 5 4.3 Introduction ............................................................................................................................................. 5 4.4 Experimental Set-up................................................................................................................................ 6 4.5
Procedure .......................................................................................................................................... 7
(a)
For a step change: ......................................................................................................................... 8
(b)
For an impulse change: ................................................................................................................. 8
4.6 Result and Discussion ............................................................................................................................. 8 a) For a step change .................................................................................................................................. 8 b) For an impulse change .......................................................................................................................... 9 4.7
Conclusion ...................................................................................................................................... 10
4.8 References ............................................................................................................................................. 10 4.9 Appendix A4 ......................................................................................................................................... 11
List of Figures Figure 1: Two tank interacting system ......................................................................................................... 5 Figure 2: Experimental Set-up ...................................................................................................................... 7 Figure 3: Response observed for a step change ............................................................................................ 9 Figure 4: Response to an impulse change ................................................................................................... 10
List of Tables Table 1: Observation Table for step change ............................................................................................... 11 Table 2: Observation for impulse change ................................................................................................... 14
3
4.1 Executive Summary The objective of the experiment is to study the dynamic response of two interacting tank system subjected to a step and impulse change. To achieve the objective two tanks each of area 66.4 cm2 was used. For a step change, the initial flowrate was set to 10LPH and the initial steady state height of tank 3 and 2 were obtained at 23cm and 14cm respectively and then a step change of 50LPH was introduced. The time constant of tank 3 and 2 were found to be 500s and 435s respectively. The observed and calculated height of tank 2 was found to be slightly different and the reason for the same may be parallax error or the non-linearity of resistance. For an impulse change the initial flowrate was set to 20LPH the initial steady state height of tank 3 and 2 were obtained at 24cm and 18cm respectively and then 500ml of water was added to 3rd tank and it was observed that the water level increases all of a sudden and then it starts to fall and reaches its initial steady state value after sometime.
4
4.2 Objective The objective of the experiment is to study the dynamic response of two interacting tank system subjected to step and impulse change
4.3 Introduction A physical system can be represented by several first-order processes connected in series [1]. One possible example for such an arrangement is shown in Fig.1.It consists of two tanks connected in an interacting manner in which two tanks are arranged such that the outlet flow from the first tank is the inlet flow to the second tank. In such a system the outlet flow from tank 1 discharges directly into tank 2, and the flow through R1 (flow resistance) depends on the difference between h1 and h2 (liquid level in tank 1 and 2 respectively).
Figure 1: Two tank interacting system [1]
Let us assume the liquid to be of constant density (Ο), the tanks to have uniform cross-sectional area (A1 and A2), and the flow resistances are R1 and R2. Using the notations mentioned on figure 1, we can write the following equations [1]: For Tank 1: πβ1 β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.1) ππ‘ β1 β β2 π 1 = β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.2) π1
π β π1 = π΄1
For Tank 2: π1 β π2 = π΄2
πβ2 β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.3) ππ‘
5
π 2 =
β2 β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.4) π2
At steady state the above equations can be written as: dh1s β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.5) dt h1s β h2s = β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.6) q1s
q s β q1s = A1 R1s
q1s β q 2s = A2 R 2s =
dh2s β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.7) dt
h2s β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ β¦ . (4.8) q 2s
On subtracting (4.1-4.5), (4.2-4.6), (4.3-4.7), (4.4-4.8), introducing deviation variables and taking Laplace Transforms we get: Q(s) β Q1 (s) = A1 (s)sH1 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ (4.9) Q1 (s) β Q2 (s) = A2 (s)sH2 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ (4.10) R1 (s)Q1 (s) = H1 (s) β H2 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.11) R 2 (s)Q2 (s) = H2 (s) β¦ β¦ β¦ β¦ β¦ β¦ β¦ . . (4.12) H2 (s) R2 = β¦ β¦ β¦ β¦ β¦ (4.13) Q(s) Ο2 Ο1 s 2 + (Ο1 + Ο2 + A1 R 2 )s + 1 The difference between the transfer function for the non-interacting system and that for the interacting system is the presence of the cross product term A1R2 in the coefficient of s.
4.4 Experimental Set-up The apparatus shown in Fig. 2 is used to study the response of a two tank non-interacting tanks in series. It consists of supply tank, pump for water circulation from the sump tank to either tank 1 or 3 depending upon the type of system to be studied, a rotameter for inlet flow measurement and 3 graduated tanks which can be connected in interacting and non-interacting mode. The
6
graduations in the tank help to measure the level of water in the tank at a particular time. For this experiment we use interacting mode i.e. tank 3 and tank 2.
Figure 2: Experimental Set-up
4.5 Procedure ο·
Ensure all drain valves are closed.
ο·
Fill water in the sump tank.
ο·
Switch on the pump and allow the water to flow to tank 3.
ο·
Set the desired flowrate of water to Tank 3 using the rotameter provided.
ο·
Adjust the valves so as to attain steady state in tank 3 and tank 2
ο·
Once steady state is achieved record the initial flowrate and the steady state level of both the tanks
7
(a) ο·
For a step change: Once the steady state is achieved, apply step up change by increasing the inlet flowrate to tank 3 using the rotameter.
ο·
Record the water level in both tanks after every fixed interval of time till steady state is reached.
ο·
Once the steady state is achieved, record the final level of both the tanks.
(b)
For an impulse change: ο·
Once the system reaches steady state apply impulse change by adding 500 ml of water directly to Tank 3 and record the first data immediately.
ο·
Record the water level in both tanks after every fixed interval of time till steady state is reached.
ο·
Once the steady state is achieved, record the final level of both the tanks.
4.6 Result and Discussion a) For a step change Two tanks each of diameter 9.2cm and area of 66.47cm2 was used in the experiment. The initial flowrate to tank 3 was maintained at 10LPH. Initial steady state level of water in tank 3 (h1s) was obtained at 22cm and initial steady state level of water in tank 2 (h2s) was found to be 14cm. A step up change of magnitude 50 was applied i.e. the flowrate was changed to 60LPH. The change in water level with respect to time, on applying step change, for both the tank is shown in Fig. 3. It also shows the calculated value of H2 and the calculation is shown in Appendix A4. The final water level obtained in tank 3 and tank 2 was 126 and 105 cm respectively. The values of Ο1 and Ο2 were calculated to be 500s and 435s respectively. The calculation is shown in Appendix A4. As it can be seen from Fig. 3 the observed and calculated value of H2(t) is not exactly same and the calculated response reaches steady state after a very long time as compared to what is observed. Deviations observed may be due to following factors: β’ Non-linearity of valve resistance. β’ Step change is not instantaneous. β’ Visual errors in recording observations. β’ Accuracy of rotameters. In terms of transient response the interacting system is more sluggish than the non-interacting system. 8
120 100
Height(cm)
80 H1
60
H2 observed
40
H2 calculated
20 0 0
100
200
300
400
500
600
700
time(s) Figure 3: Response observed for a step change
b) For an impulse change Two tanks each of diameter 9.2cm and area of 66.47cm2 was used in the experiment. The initial flowrate to tank 3 was maintained at 20LPH. Initial steady state level of water in tank 3 (h1s) was obtained at 24cm and initial steady state level of water in tank 2 (h2s) was found to be 18cm. For an impulse change 500cm3 of water was added to tank 3 quickly at the water level in tank 3 and 2 at t=0 was found to be 94cm and 42cm respectively. The change in water level with respect to time, on applying impulse change, for both the tank is shown in Fig. 4. The final water level obtained in tank 3 and tank 2 was 24cm and 18 cm respectively. It can be seen from Fig. 4 that the output for an impulse change first increases as soon as the impulse is applied and then decreases and reaches a value which is equal to the initial height. The curves obtained in Fig. 4 are not smooth. Deviations observed may be due to following factors: β’ Non-linearity of valve resistance. β’ Step change is not instantaneous. β’ Visual errors in recording observations. β’ Accuracy of rotameters.
9
80 70 60
height(cm)
50 40
H1
30
H2
20 10 0 0
50
100
150
200
250
300
350
400
450
-10 time(s) Figure 4: Response to an impulse change
4.7 Conclusion The objective of the experiment was achieved by using two cylindrical tanks of equal areas (66.4 cm2). For a step change of 50LPH in the flowrate the values of Ο1 and Ο2 obtained were 500s and 435s respectively. For an interacting tank system the flow through R1 depends on the difference in the water level in the two tanks. .For an impulse the water level first increases and then decreases and comes back to its original after sometime. In terms of transient response the interacting system is more sluggish than the non-interacting system. Any error that might have occurred may be due to parallax or non-linearity of valve resistance. Hence, for more accurate results, it is recommended that the experiment should be allowed to run for an hour before step or impulse change is introduced. This would ensure that steady state is achieved and hence would result in much more precise results.
4.8 References 1. D. Coughanowr and S. Le Blanc, Process System Analysis and Control, Mc-Graw Hill (1991)(71-92).
10
4.9 Appendix A4 Table 1: Observation Table for step change
time
h1
h2
H1
H2(observed) H2(calculated)
(s)
(cm)
(cm)
(cm)
(cm)
(cm)
0
22
14
0
0
0
10
32
15
10
1
0.07377779
20
37
15
15
1
0.289038299
30
41
15
19
1
0.637043998
40
46
15
24
1
1.109531867
50
49
16
27
2
1.698688095
60
53
16
31
2
2.397124126
70
56
17
34
3
3.197853975
80
60
18
38
4
4.094272759
90
64
19
42
5
5.080136361
100
66
20
44
6
6.149542177
110
70
21
48
7
7.296910889
120
72
22
50
8
8.516969206
130
75
23
53
9
9.804733522
140
78
24
56
10
11.15549445
150
82
26
60
12
12.56480215
160
84
27
62
13
14.0284525
170
86
28
64
14
15.54247392
180
88
30
66
16
17.10311496
190
90
32
68
18
18.70683254
200
92
34
70
20
20.35028076
210
94
36
72
22
22.03030039
220
96
37
74
23
23.74390885
230
98
39
76
25
25.48829078
240
100
41
78
27
27.26078905
11
250
101
43
79
29
29.05889628
260
102
45
80
31
30.88024684
270
103
48
81
34
32.72260923
280
104
50
82
36
34.58387886
290
105
53
83
39
36.46207128
300
106
55
84
41
38.3553157
310
107
57
85
43
40.26184889
320
108
60
86
46
42.18000937
330
109
62
87
48
44.10823198
340
110
64
88
50
46.04504267
350
111
66
89
52
47.9890536
360
112
67
90
53
49.93895846
370
113
68
91
54
51.89352815
380
114
69
92
55
53.8516065
390
115
70
93
56
55.81210646
400
116
71
94
57
57.77400624
410
117
73
95
59
59.73634587
420
118
75
96
61
61.6982238
430
119
77
97
63
63.65879376
440
120
79
98
65
65.61726176
450
120
81
98
67
67.57288322
460
120
83
98
69
69.52496032
470
120
86
98
72
71.47283945
480
120
89
98
75
73.41590878
490
121
92
99
78
75.353596
500
121
94
99
80
77.28536614
510
121
95
99
81
79.21071958
520
121
96
99
82
81.12919003
530
122
97
100
83
83.04034277
540
122
98
100
84
84.94377291
12
550
123
99
101
85
86.83910371
560
123
100
101
86
88.72598505
570
124
101
102
87
90.60409197
580
124
103
102
89
92.47312328
590
125
103
103
89
94.3328002
600
125
104
103
90
96.18286518
610
126
104
104
90
98.02308067
620
126
104
104
90
99.85322801
630
126
105
104
91
101.6731064
640
126
105
104
91
103.4825319
650
126
105
104
91
105.2813364
660
126
105
104
91
107.0693668
Calculation: For time constant of Tank 1 dQ1 = 60 - 10 = 50 LPH dH1 = 126 - 22 = 104cm R1 = dH1/ dQ1 = 7.488 s/cm2 Ο1 = A1*R1 = 500s For calculated height of tank 2: π=
1 1 π΄1 π 2 + + = 0.006302 π1 π2 π1 π2
π π 1 πΌ = (β ) + β( )2 β = β0.00084 2 2 π1 π2 π π 1 π½ = (β ) β β( )2 β = β0.00546 2 2 π1 π2 1 1 [πΌ π πΌπ‘ ] β π π½π‘ π½ π»2 (π‘)πππππ’πππ‘ππ = π΄π 2 {1 β } 1 1 πΌβπ½ At t=10 secs H2=0.074 cm. 13
Table 2: Observation for impulse change
time
h1
h2
H1
H2
Steady
24
18
0
0
0
94
42
70
24
10
74
44
50
26
15
70
46
46
28
20
66
48
42
30
30
60
50
36
32
40
58
48
34
30
50
57
46
33
28
60
56
44
32
26
70
54
42
30
24
80
52
40
28
22
90
50
38
26
20
100
49
36
25
18
110
47
34
23
16
120
46
33
22
15
130
45
32
21
14
140
44
31
20
13
150
43
30
19
12
160
42
30
18
12
170
41
29
17
11
180
40
28
16
10
190
39
27
15
9
200
38
26
14
8
210
37
25
13
7
220
36
25
12
7
230
35
24
11
6
240
34
23
10
5
state
14
250
33
22
9
4
260
32
21
8
3
270
31
20
7
2
280
30
20
6
2
290
30
19
6
1
300
29
19
5
1
310
28
18
4
0
320
28
18
4
0
330
27
18
3
0
340
27
18
3
0
350
26
18
2
0
360
26
18
2
0
370
25
18
1
0
380
25
18
1
0
390
24
18
0
0
400
24
18
0
0
410
24
18
0
0
420
24
18
0
0
15
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